A short paper with references to several early counterexamples proves that (in the good category of compactly generated weak Hausdorff spaces) a Serre fibration in which the total space and base space are both CW complexes is necessarily a Hurewicz fibration.  M. Steinberger and J. West. Covering homotopy properties of maps between CW complexes or ANRs. Proc. Amer. Math. Soc. 92(1984), 573-577.  (The proof is corrected in R. Cauty. Sur les overts des CW-complexes et les fibr\'es de Serre.  Colloquy Math. 63(1992), 1--7).  No relationship between the covering map and the CW structures is required.  This argues either that counterexamples are 
pathological or that it is a special property for the total space of a Serre fibration with CW base space to be a CW complex, although it has the homotopy type of a CW complex if the fiber does.